TY - CHAP A1 - Pfetsch, Marc A1 - Schmidt, Martin A1 - Skutella, Martin A1 - Thürauf, Johannes T1 - Potential-Based Flows - An Overview T2 - Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks N2 - Potential-based flows provide an algebraic way to model static physical flows in networks, for example, in gas, water, and lossless DC power networks. The flow on an arc in the network depends on the difference of the potentials at its end-nodes, possibly in a nonlinear way. Potential-based flows have several nice properties like uniqueness and acyclicity. The goal of this paper is to provide an overview of the current knowledge on these models with a focus on optimization problems on such networks. We cover basic properties, computational complexity, monotonicity, uncertain parameters, and the corresponding behavior of the network as well as topology optimization. Y1 - 2026 PB - Springer-Nature ER - TY - CHAP A1 - Leugering, Günter T1 - Nonoverlapping Domain Decomposition in Space and Time for Optimal Control Problems on Metric Graphs by the Example of Gas Flow in Pipe Networks N2 - We consider non-overlapping domain decomposition methods for ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we chose a semilinear approximation of the Euler system and a doubly nonlinear parabolic model that has come to be known as friction dominated flow in gas pipe networks. By this choice, we encounter hyperbolic, parabolic and elliptic linear and nonlinear problems in a single highly motivating application. We depart from the classical domain decomposition methods described by P.L. Lions and J.L. Lions and O. Pironneau and extend those to problems on metric graphs. The choice of methods is determined by the desire to use a control concept that has come to be known as virtual controls which, in turn, possibly lead to a fully parallel decomposition of the corresponding optimality systems. In a second step, we extend the methods to p-Laplace problems on networks. The analysis, due to space limitations, will appear in a forthcoming publication. See however J.E. Lagnese and G. Leugering. Furthermore, we then describe methods for space and time domain decomposition or optimal control problems in the spirit of J.E. Lagnese and G. Leugering. We finally provide some comments on PINN-based approximations of the methods described before. We provide numerical evidence for all algorithms discussed. KW - Optimal control KW - PDEs on graphs KW - p-Laplace problem on a graph KW - p-parabolic problems KW - instantaneous control Y1 - 2022 ER - TY - CHAP A1 - Benner, Peter A1 - Grundel, Sarah A1 - Himpe, Christian A1 - Huck, Christoph A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - Gas Network Benchmark Models T2 - Applications of Differential-Algebraic Equations: Examples and Benchmarks N2 - The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models. Y1 - 2017 U6 - https://doi.org/10.1007/11221_2018_5 VL - DAE-FORUM SP - 171 EP - 197 PB - Springer CY - Cham ER -